Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees

Ryota Inagaki, Tanya Khovanova, Austin Luo

TL;DR

The paper studies how permutation-based firing strategies on infinite directed k-ary trees produce distinct stable configurations in the labeled chip-firing model. By encoding strategies with permutations w in S_n and analyzing chip trajectories via Lehmer codes, the authors derive explicit formulas for inversions and descents, show monotonicity under a domination order, and identify structural patterns such as a lexicographic correspondence mediated by a B map. They also explore tails of permutations (valleys) to obtain tight bounds and demonstrate nested growth of inversion values, including a limiting sequence. The work provides a precise, combinatorial framework linking permutation structure to stable configurations, with potential implications for understanding the space of reachable configurations under deterministic strategies on trees.

Abstract

Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed $k$-ary tree where we place $k^n$ chips labeled $0,1,\dots, k^n-1$ on the root for some nonnegative integer $n$, and we say a vertex $v$ can fire if it has at least $k$ chips. When a vertex fires, we select $k$ labeled chips and send the $i$th smallest chip among them to its $i$th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of $1, 2, \dots, n$. We then express the stable configuration as a permutation of $0,1, 2, \dots, k^n-1$ and explore its properties, such as the number of inversions and descents.

Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees

TL;DR

The paper studies how permutation-based firing strategies on infinite directed k-ary trees produce distinct stable configurations in the labeled chip-firing model. By encoding strategies with permutations w in S_n and analyzing chip trajectories via Lehmer codes, the authors derive explicit formulas for inversions and descents, show monotonicity under a domination order, and identify structural patterns such as a lexicographic correspondence mediated by a B map. They also explore tails of permutations (valleys) to obtain tight bounds and demonstrate nested growth of inversion values, including a limiting sequence. The work provides a precise, combinatorial framework linking permutation structure to stable configurations, with potential implications for understanding the space of reachable configurations under deterministic strategies on trees.

Abstract

Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed -ary tree where we place chips labeled on the root for some nonnegative integer , and we say a vertex can fire if it has at least chips. When a vertex fires, we select labeled chips and send the th smallest chip among them to its th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of . We then express the stable configuration as a permutation of and explore its properties, such as the number of inversions and descents.

Paper Structure

This paper contains 19 sections, 18 theorems, 25 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unlabeled chip-firing on directed graphs
  3. Labeled chip-firing on directed graphs
  4. Motivation
  5. Our Objective and Main Definition
  6. Roadmap
  7. Preliminaries and Definitions
  8. Permutations
  9. The underlying graph
  10. Unlabeled chip-firing on directed k-ary trees
  11. Labeled chip-firing on directed k-ary trees
  12. Permutation-Based Strategies of Chip-Firing
  13. Inversions in Resulting Configurations
  14. The number of inversions
  15. Properties of the number of inversions
  16. ...and 4 more sections

Key Result

Theorem 1.1

For a directed graph $G$ and initial configuration $\mathcal{C}$ of chips on the graph, the unlabeled chip-firing game will either run forever or end after the same number of moves and at the same stable configuration. Furthermore, the number of times each vertex fires is the same regardless of the

Figures (4)

  • Figure 1: Example of unlabeled chip-firing on an infinite directed, rooted binary tree
  • Figure 2: Example of confluence breaking
  • Figure 3: Example of firing with strategy $F_{132}$
  • Figure 4: Example of labeled chip-firing in a directed binary tree with $4$ chips

Theorems & Definitions (61)

  • Example 1
  • Theorem 1.1: Theorem 1.1 of MR1203679
  • Example 2
  • Definition 1
  • Example 3
  • Example 4
  • Example 5
  • Definition 2
  • Proposition 3.1
  • proof
  • ...and 51 more